The problem of separating two or more signals that are superimposed on each other arise in many signal processing applications. A classical example is the problem of enhancing a desired speech signal in the presence of interference, or separating competing speakers using multiple microphone measurements. Another example is the problem of detecting a signal in the presence of directional interference, or detecting and localizing several spatially separated sources, by simultaneous observation of their signals using a spatially distributed array of receivers. Since the source signals as well as the transmission channel that couples the signals to the receivers are generally unknown, the problem of decoupling and identifying the two can not be solved unless we make some simplifying assumptions. In many cases of interest the channel can be modeled as a stable linear time invariant (LTI) system, and since the signals are generated by different sources we may also assume that they are statistically independent. The question is then whether these assumptions are sufficient for separating and recovering the source signals.
A recent series of papers (1)--C. Jutten, and J. Herault, "Blind separation of sources, part I: An adaptive algorithm based on neuromimetic architecture," Signal Processing, vol 24, No. 1, pp 1-10, July 1991, (2)--P. Comon, C. Jutten, and J. Herault, "Blind separation of sources, part II: Problems statement," Signal Processing, vol 24, No. 1, pp 11-20, July 1991, and (3)--E. Sorouchyari, C. Jutten, and J. Herault, "Blind separation of sources, part III: Stability analysis," Signal Processing, vol 24, No. 1, pp 21-29, July 1991, consider the problem of source separation for the special case in which the observed signals at each point in time are given by unknown but fixed linear combinations of the source signals (i.e. the unknown channel is a constant gain matrix). These papers contains several important observations.
First, it is pointed out that for the purpose of source separation it may be sufficient to assume that the signals are statistically independent. No additional information concerning the stochastic nature of each of the sources signals is required. To separate the sources it is then suggested to try and make the recovered signals statistically independent. However, tests for statistical independence are generally complicated, and require prior assumptions concerning the underlying probability distribution of each of the signals. Therefore, to solve the problem it is suggested to use an objective function consisting of high order statistics of the reconstructed signals, and then apply an algorithm to minimize that objective. If the algorithm converges, then at the point of convergence it attempts to minimize the correlation between some pre-specified non-linear functions of the reconstructed signals computed at the same sampling instance. Lack of correlation between non-linear functions of samples of the reconstructed processes is a weaker condition than statistical independence between samples of the processes, which is a weaker condition than statistical independence between the entire processes. However, because of the simplified channel model, it is a sufficient condition for signal separation. In fact it can be shown that decorrelating the reconstructed signals at the same sampling time is also sufficient in this case (not as stated in paper (1) above).
Patent application Ser. No. 07/750,917, filed Aug. 28, 1991 by E. Weinstein (one of the joint inventors in the present application), M. Feder and A. V. Oppenheim, for "Multi-Channel Signal Separation", assigned to Massachusetts Institute of Technology, (unpublished), considers the more general problem in which the channel is modeled as multi-input-multi-output (MIMO) LTI system. In the above patent application, it is suggested to separate and recover the input (source) signals by imposing the condition that the reconstructed signals be statistically uncorrelated. As it turns out, the decorrelation condition is insufficient to uniquely solve the problem, unless we make certain assumptions concerning the unknown channel, e.g. that it is a MIMO finite impulse response (FIR) filter.
For simplicity we shall concentrate specifically in the two channel case, in which we observe the outputs of an unknown 2.times.2 system, from which we want to recover its input signals. However we note that all the analysis and results can easily be extended to the more general case. In the two channel case it is desired to estimate the two source signals s.sub.1 [n] and s.sub.2 [n], from the observation of the two output signals y.sub.1 [n] and y.sub.2 [n]. In some applications one of the signals, s.sub.1 [n] is the desired signal, while the other signal, s.sub.2 [n], is the interference or noise signal. Both the desired and the noise signals are coupled through an unknown system to form the observed signals.